Broer-Kaup-Kupershmidt族的非线性双可积耦合及其自相容源
|
摘要
基于新的非半单矩阵李代数,介绍了构造孤子族非线性双可积耦合的方法,由相应的变分恒等式给出了孤子族非线性双可积耦合的Hamilton结构.作为应用,给出了Broer-Kaup-Kupershmidt族的非线性双可积耦合及其Hamilton结构.最后指出了文献中的一些错误,利用源生成理论建立了新的公式,并导出了带自相容源Broer-Kaup-Kupershmidt族的非线性双可积耦合方程.
Based on new non-semisimple matrix Lie algebras, the general method of constructing the nonlinear bi-integrable couplings of soliton hierarchy is introduced. The corresponding variational identity yields Hamiltonian structures of the resulting bi-integrable couplings. As an application, the nonlinear bi-integrable couplings of Broer-Kaup-Kupershmidt hierarchy and their Hamiltonian structures are given. Finally, some errors exist in reference are pointed out, and a set of new formulae using the theory of source are set up, also the nonlinear bi-integrable couplings of Broer-Kaup-Kupershmidt hierarchy with self-consistent sources is derived based on the new formulae.
Based on new non-semisimple matrix Lie algebras, the general method of constructing the nonlinear bi-integrable couplings of soliton hierarchy is introduced. The corresponding variational identity yields Hamiltonian structures of the resulting bi-integrable couplings. As an application, the nonlinear bi-integrable couplings of Broer-Kaup-Kupershmidt hierarchy and their Hamiltonian structures are given. Finally, some errors exist in reference are pointed out, and a set of new formulae using the theory of source are set up, also the nonlinear bi-integrable couplings of Broer-Kaup-Kupershmidt hierarchy with self-consistent sources is derived based on the new formulae.
引文
[1]Ma Wenxiu,Fushssteiner B.The bi-Hamiltonian structures of the perturbation equations of Kd V hierarchy[J].Phys Lett A,1996,213(1):49-55.
[2]Ma Wenxiu,Fushssteiner B.Integrable theory of the perturbation equations[J].Chaos Soliton Fract,1996,7(8):1227-1250.
[3]Ma Wenxiu.Enlarging spectral problems to construct integrable couplings of soliton equations[J].Phys Lett A,2003,316(1):72-76.
[4]Ma Wenxiu,Xu Xixiang,Zhang Yufeng.Semi-direct sums of Lie algebras and continuous integrable couplings[J].Phys Lett A,2006,351(3):125-130.
[5]Zhang Yufeng,Zhang Hongqing.A direct method for integrable couplings of TD hierarchy[J].J Math Phys,2002,43(1):466-472.
[6]Zhang Yufeng.A generalized multi-component Glachette-Johnson(GJ)hierarchy and its integrable coupling system[J].Chaos Soliton Fract,2004,21(2):305-310.
[7]Zhang Yufeng,Yan Qingyou.Integrable couplings of the hierarchies of evolution equations[J].Chaos Soliton Fract,2003,16(2):263-269.
[8]Xia Tiecheng,Fan Engui.The multicomponent generalized Kaup-Newell hierarchy and its multicomponent integrable couplings system with two arbitrary functions[J].J Math Phys,2005,46(4):043510.
[9]You Fucai,Xia Tiecheng.Integrable couplings of the generalized AKNS hierarchy with an arbitrary function and its bi-Hamiltonian structure[J].Int J Theoret Phys,2007,46(12):3159-3168.
[10]Zhang Yufeng,Guo Fukui.Two pairs of Lie algebras and the integrable couplings as well as the Hamiltonian structure of the Yang hierarchy[J].Chaos Soliton Fract,2007,34(2):490-495.
[11]Wei Hanyu,Xia Tiechneg.Generalized fractional trace variational identity and a new fractional integrable couplings of soliton hierarchy[J].Acta Math Sci,2014,34B(1):53-64.
[12]Xia Tiecheng.Integrable couplings of classical-Boussinesq hierarchy and its Hamiltonian structure[J].Chin Phys B,2010,53(1):25-27
[13]Ma Wenxiu.Nonlinear continuous integrable Hamiltonian couplings[J].Appl Math Comput,2011,217(17):7238-7244.
[14]Zhang Yufeng,Mei Jianqin.Lie Algebras and integrable systems[J].Commun Theor Phys,2012,57(6):1012-1022.
[15]Zhang Yufeng.Lie Algebras for constructing nonlinear integrable couplings[J].Commun Theor Phys,2011,56(5):805-812.
[16]You Fucai.Nonlinear super integrable Hamiltonian couplings[J].J Math Phys,2011,52(12):123510.
[17]Wang Hui,Xia Tiecheng.Three nonlinear integrable couplings of the nonlinear Schrodinger equations[J].Commun Nonlinear Sci Numer Simul,2011,16(11):4232-4237.
[18]Yu Fajun,Li li.A nonlinear integrable couplings of C-Kd V soliton hierarchy and its infinite conservation laws[J].Appl Math Comput,2014,233(1):413-420.
[19]Wei Hanyu,Xia Tiecheng.The generalized fractional trace variational identity and fractional integrable couplings of Kaup-Newell hierarchy[J].J Math Phys,2014,55(8):083501.
[20]Tu Guizhang.The trace identity,a powerful tool for constructing the Hamiltonian structure of integrable systems[J].J Math Phys,1989,30(2):330-338.
[21]Ma Wenxiu.A new hierarchy of Liouville integrable generalized Hamiltonian equations and its reduction[J].Chin Ann Math,1992,13(1):115-123.
[22]Guo fukui,Zhang yufeng.The quadratic-form identity for constructing the Hamiltonian structure of integrable systems[J].J Phys A:Mathe Gen,2005,38(40):8537-8548.
[23]Ma Wenxiu.Soliton,positon and negaton solutions to a Schroedinger self-consistent source equation[J].J Phys Soc Japan,2003,72(12):3017-3019.
[24]Wang hui,Xia Tiecheng.Conservation laws for a super G-J hierarchy with self-consistent sources[J].Commun Nonlinear Sci Numer Simul,2012,17(2):566-572.
[25]Li Li.Conservation laws and self-consistent sources for a super-CKd V equation hierarchy[J].Phys Lett A,2011,375(11):1402-1406.
[26]Xia Tiecheng.Two new integrable couplings of the soliton hierarchies with self-consistent sources[J].Chin Phys B,2010,19(10):100303
[27]Mel’nikov V K.Integration of the Korteweg-de Vries equation with a source[J].Inverse Probl,1990,6(2):233-246.
[28]Zakharov V E,Kuznetsov E A.Multi-scale expansions in the theory of systems integrable by the inverse scattering transform[J].Phys D,1986,18(3):455-463.
[29]Yu Fajun,Li Li.Integrable coupling system of JM equations hierarchy with self-consistent sources[J].Commun Theor Phys,2010,53(1):6-12.
[30]Ma Wenxiu,Chen Min.Hamiltonian and quasi-Hamiltonian structures associated with semi-direct sums of Lie algebras[J].J Math Phys,2006,39(34):10787-10801.
[31]Ma Wenxiu,Zhang Yi.Component-trace identities for Hamiltonian structures[J].Appl Anal,2010,89(4):457-472.
[32]Yang Hongxiang,Sun Yepeng.Hamiltonian and super-hamiltonian extensions related to Broer-Kaup-Kupershmidt system[J].Int J Theoret Physics,2009,49(2):349-364.
[33]Ma Wenxiu,Meng Jinghan,Zhang Mengshu.Nonlinear bi-integrable couplings with Hamiltonian structures[J].Math Comput Simul,2016,127:166-177.
[34]Tu Guizhang.An extension of a theoremon gradients of conserved densities of integrable systems[J].Northeast Math J,1990,6(1):26-32.
[2]Ma Wenxiu,Fushssteiner B.Integrable theory of the perturbation equations[J].Chaos Soliton Fract,1996,7(8):1227-1250.
[3]Ma Wenxiu.Enlarging spectral problems to construct integrable couplings of soliton equations[J].Phys Lett A,2003,316(1):72-76.
[4]Ma Wenxiu,Xu Xixiang,Zhang Yufeng.Semi-direct sums of Lie algebras and continuous integrable couplings[J].Phys Lett A,2006,351(3):125-130.
[5]Zhang Yufeng,Zhang Hongqing.A direct method for integrable couplings of TD hierarchy[J].J Math Phys,2002,43(1):466-472.
[6]Zhang Yufeng.A generalized multi-component Glachette-Johnson(GJ)hierarchy and its integrable coupling system[J].Chaos Soliton Fract,2004,21(2):305-310.
[7]Zhang Yufeng,Yan Qingyou.Integrable couplings of the hierarchies of evolution equations[J].Chaos Soliton Fract,2003,16(2):263-269.
[8]Xia Tiecheng,Fan Engui.The multicomponent generalized Kaup-Newell hierarchy and its multicomponent integrable couplings system with two arbitrary functions[J].J Math Phys,2005,46(4):043510.
[9]You Fucai,Xia Tiecheng.Integrable couplings of the generalized AKNS hierarchy with an arbitrary function and its bi-Hamiltonian structure[J].Int J Theoret Phys,2007,46(12):3159-3168.
[10]Zhang Yufeng,Guo Fukui.Two pairs of Lie algebras and the integrable couplings as well as the Hamiltonian structure of the Yang hierarchy[J].Chaos Soliton Fract,2007,34(2):490-495.
[11]Wei Hanyu,Xia Tiechneg.Generalized fractional trace variational identity and a new fractional integrable couplings of soliton hierarchy[J].Acta Math Sci,2014,34B(1):53-64.
[12]Xia Tiecheng.Integrable couplings of classical-Boussinesq hierarchy and its Hamiltonian structure[J].Chin Phys B,2010,53(1):25-27
[13]Ma Wenxiu.Nonlinear continuous integrable Hamiltonian couplings[J].Appl Math Comput,2011,217(17):7238-7244.
[14]Zhang Yufeng,Mei Jianqin.Lie Algebras and integrable systems[J].Commun Theor Phys,2012,57(6):1012-1022.
[15]Zhang Yufeng.Lie Algebras for constructing nonlinear integrable couplings[J].Commun Theor Phys,2011,56(5):805-812.
[16]You Fucai.Nonlinear super integrable Hamiltonian couplings[J].J Math Phys,2011,52(12):123510.
[17]Wang Hui,Xia Tiecheng.Three nonlinear integrable couplings of the nonlinear Schrodinger equations[J].Commun Nonlinear Sci Numer Simul,2011,16(11):4232-4237.
[18]Yu Fajun,Li li.A nonlinear integrable couplings of C-Kd V soliton hierarchy and its infinite conservation laws[J].Appl Math Comput,2014,233(1):413-420.
[19]Wei Hanyu,Xia Tiecheng.The generalized fractional trace variational identity and fractional integrable couplings of Kaup-Newell hierarchy[J].J Math Phys,2014,55(8):083501.
[20]Tu Guizhang.The trace identity,a powerful tool for constructing the Hamiltonian structure of integrable systems[J].J Math Phys,1989,30(2):330-338.
[21]Ma Wenxiu.A new hierarchy of Liouville integrable generalized Hamiltonian equations and its reduction[J].Chin Ann Math,1992,13(1):115-123.
[22]Guo fukui,Zhang yufeng.The quadratic-form identity for constructing the Hamiltonian structure of integrable systems[J].J Phys A:Mathe Gen,2005,38(40):8537-8548.
[23]Ma Wenxiu.Soliton,positon and negaton solutions to a Schroedinger self-consistent source equation[J].J Phys Soc Japan,2003,72(12):3017-3019.
[24]Wang hui,Xia Tiecheng.Conservation laws for a super G-J hierarchy with self-consistent sources[J].Commun Nonlinear Sci Numer Simul,2012,17(2):566-572.
[25]Li Li.Conservation laws and self-consistent sources for a super-CKd V equation hierarchy[J].Phys Lett A,2011,375(11):1402-1406.
[26]Xia Tiecheng.Two new integrable couplings of the soliton hierarchies with self-consistent sources[J].Chin Phys B,2010,19(10):100303
[27]Mel’nikov V K.Integration of the Korteweg-de Vries equation with a source[J].Inverse Probl,1990,6(2):233-246.
[28]Zakharov V E,Kuznetsov E A.Multi-scale expansions in the theory of systems integrable by the inverse scattering transform[J].Phys D,1986,18(3):455-463.
[29]Yu Fajun,Li Li.Integrable coupling system of JM equations hierarchy with self-consistent sources[J].Commun Theor Phys,2010,53(1):6-12.
[30]Ma Wenxiu,Chen Min.Hamiltonian and quasi-Hamiltonian structures associated with semi-direct sums of Lie algebras[J].J Math Phys,2006,39(34):10787-10801.
[31]Ma Wenxiu,Zhang Yi.Component-trace identities for Hamiltonian structures[J].Appl Anal,2010,89(4):457-472.
[32]Yang Hongxiang,Sun Yepeng.Hamiltonian and super-hamiltonian extensions related to Broer-Kaup-Kupershmidt system[J].Int J Theoret Physics,2009,49(2):349-364.
[33]Ma Wenxiu,Meng Jinghan,Zhang Mengshu.Nonlinear bi-integrable couplings with Hamiltonian structures[J].Math Comput Simul,2016,127:166-177.
[34]Tu Guizhang.An extension of a theoremon gradients of conserved densities of integrable systems[J].Northeast Math J,1990,6(1):26-32.